Problem: The water tank in the diagram below is in the shape of an inverted right circular cone. The radius of its base is 16 feet, and its height is 96 feet. The water in the tank is $25\%$ of the tank's capacity.  The height of the water in the tank can be written in the form $a\sqrt[3]{b}$, where $a$ and $b$ are positive integers and $b$ is not divisible by a perfect cube greater than 1.  What is $a+b$?

[asy]
size(150);
defaultpen(linewidth(.8pt)+fontsize(8pt));

draw(shift(0,96)*yscale(0.5)*Circle((0,0),16));
draw((-16,96)--(0,0)--(16,96)--(0,96));

draw(scale(0.75)*shift(0,96)*yscale(0.5)*Circle((0,0),16));

draw((-18,72)--(-20,72)--(-20,0)--(-18,0));
label("water's height",(-20,36),W);

draw((20,96)--(22,96)--(22,0)--(20,0));
label("96'",(22,48),E);

label("16'",(8,96),S);
[/asy]
Solution: The water in the tank fills a cone, which we will refer to as the water cone, that is similar to the cone-shaped tank itself.  Let the scale factor between the water cone and tank be $x$, so the height of the water cone is $96x$ feet and the radius of the water cone is $16x$ feet.  It follows that the volume of the water cone is $(1/3)\pi(16x)^2(96x)$ cubic feet.

The volume of the cone-shaped tank is $(1/3)\pi(16^2)(96)$.  Since the water cone has $25\%$ or 1/4 of the volume of the tank, we have  \[(1/3)\pi(16x)^2(96x) = (1/4) (1/3)\pi(16^2)(96).\]  Simplifying yields $x^3 = 1/4$, so $x = \sqrt[3]{1/4}$.

Finally, the height of the water in the tank is the height of the water cone, which is  \[96x=96\sqrt[3]{1/4}=48\cdot 2\sqrt[3]{1/4}=48\sqrt[3]{(1/4)(8)}={48\sqrt[3]{2}}\] feet.  Therefore, we have $a+b=48+2 = \boxed{50}$.